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SimpleTrack:Adaptive Trajectory Compression with Deterministic Projection Matrix for Mobile Sensor Networks
arXiv:1404.6151v1 [cs.IT] 23 Apr 2014
Rajib Rana, Member, IEEE, Mingrui Yang, Member, IEEE, Tim Wark, Senior Member, IEEE, Chun Tung Chou, Member, IEEE, and Wen Hu, Senior Member, IEEE
Abstract—Some mobile sensor network applications require the sensor nodes to transfer their trajectories to a data sink. This paper proposes an adaptive trajectory (lossy) compression algorithm based on compressive sensing. The algorithm has two innovative elements. First, we propose a method to compute a deterministic projection matrix from a learnt dictionary. Second, we propose a method for the mobile nodes to adaptively predict the number of projections needed based on the speed of the mobile nodes. Extensive evaluation of the proposed algorithm using 6 datasets shows that our proposed algorithm can achieve sub-metre accuracy. In addition, our method of computing projection matrices outperforms two existing methods. Finally, comparison of our algorithm against a state-of-the-art trajectory compression algorithm show that our algorithm can reduce the error by 10-60 cm for the same compression ratio. Index Terms—Mobile sensor networks; trajectory compression; compressive sensing; adaptive compression; support vector regression; sparse coding; singular value decomposition.
I. I NTRODUCTION Mobile sensor networks (MSNs), which consists of autonomous embedded sensor nodes roaming freely, offer many new opportunities that are not available to their stationary counterparts. The Virtual Fencing (VF) [1] project that is being conducted out in our laboratory is one such example. A cattle farm generally covers an enormous area and it is costly to build fences around it. VF offers an alternative where no physical fencing is needed. The cattle carry an embedded device with GPS on board. The device constantly monitors the cow’s location and if a cow tries to leave the farm, the device sends a stimulus (either an auditory or mild electric shock) to signal the cow to turn back. Ethical considerations are critical to the VF application. Locations of the animals and records of stimuli applied must be kept. This requires the device to record the trajectory of the animal. Given the limited storage capacity of the embedded device, these stored trajectories must be uploaded to some server at some time. Due to the enormous size of the farm, base stations can only be installed at certain places. Therefore VF operates as a delay tolerant network. When a cow is getting close to a base station, the device on the animal makes use of the short transmission opportunity R. Rana is with the Department of Computation Informatics, CSIRO (The Commonwealth Scientific and Industrial Research Organisation), Australia. e-mail:
[email protected]. M. Yang, T. Wark and W. Hu are with the Department of Computation Informatics, CSIRO. Chun Tung Chou is with the School of Computer Science and Engineering, UNSW, Sydney, Australia.
available to transfer the stored trajectory to the server. Given this limited transmission opportunity, as well as limited storage on the device, trajectory compression is important. Data compression is a richly researched field, with many well known algorithms such as Lempel-Ziv [2] and many others. However, in the context of trajectory compression in MSNs, we also need to take the limited computation and transmission resources of MSN nodes into consideration. This demands for us to look for simple compression algorithm with good space savings. The recently developed theory of Compressive Sensing (CS) [3], [4] offers such a possibility because its compression step is very simple. In fact, we demonstrated in our earlier work [5] that such type of compression is feasible on an 8-bit Atmel Amega 1281 microcontroller with 8 kB RAM. However, we could only achieve an accuracy in the order of metres for the reconstructed trajectories in [5]. In this paper, we propose an improved compression scheme SimpleTrack which achieves a sub-metre accuracy. SimpleTrack is also based on CS. Given a n-dimensional data vector x to be compressed, SimpleTrack uses a m × n projection matrix Φ to compute compressed data vector y = Φx. This projection matrix Φ is fat (which means m < n), therefore the output of the compression step y has a lower dimension compared with the original data vector x. The number of projections m determines the size of the compressed data y. A smaller m means lower storage and transmission requirement. The compressed vector y is used during decompression to reconstruct an approximation of x. This reconstruction step requires the projection matrix Φ as well as a reconstruction basis Ψ. In order to achieve accurate reconstruction with low resource consumption, we need to make good choices of the three parameters: number of projections m, projection matrix Φ and reconstruction basis Ψ. In this paper, we make the following contributions: • We propose an adaptive method, based on support vector regression (SVR) [6], to enable the MSN nodes to dynamically choose the number of projections m based on the their speed. • We show that better reconstruction accuracy can be achieved by using a learnt dictionary together with a projection matrix computed from the dictionary. • We propose a new method to compute projection matrix from a dictionary. Experimental results show that our proposed method outperforms two existing methods. We also provide an explanation of why our proposed method
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works better. We perform extensive evaluation by using 6 datasets. The results show that our method is 10-60cm more accurate than the state-of-the-art trajectory compression algorithm SQUISH [7]. This paper is organized as follows. Section II presents the background on CS and dictionary learning. We describe our proposed trajectory compression algorithm SimpleTrack in Section III. Evaluations of SimpleTrack are presented in Section IV. Section V discusses related work and Section VI concludes the paper. •
II. BACKGROUND SimpleTrack uses CS and dictionary learning. We present an overview of these two topics in this section. A. Compressive Sensing (CS) CS has received a lot of attention in the past decade because it can significantly reduce the sampling requirement in many applications ([8], [9], [10]). CS has also been used in wireless sensor networks (WSNs) for reducing the energy consumption in data gathering ([11], [12], [13], [14], [15]) and the computational requirements on sensors ([16], [17], [18]). We will review the aspects of CS which are necessary for understanding this paper, more details can be found in ([4]). Given a vector x ∈ Rn , we can compute its representation θ ∈ Rn in a basis Ψ ∈ Rn×n by solving the linear equation x = Ψθ
(1)
The representation θ is said to be compressible if θ has a large number of elements with small magnitude. We can realise compression by setting these elements with small magnitude to zero. This can reduce the storage requirements. If we can find a basis in which a given vector x has a compressible representation, we will also say that x is compressible. The theory of CS applies to compressible vectors. CS considers the problem of recovering an unknown compressible vector x from its projections. Let Φ be a m × n projection matrix with m < n. Consider the equation: y = Φx + z
(2)
where z ∈ Rn is a noise vector whose norm is bounded by . CS aims to recover (or reconstruct) x from y and Φ given the knowledge that x is compressible in the basis Ψ. CS shows that under certain conditions it is possible to recover x by solving the following `1 optimisation problem: ˆ 1 min kθk
ˆ n θ∈R
ˆ 2 ≤ . subject to ky − ΦΨθk
(3)
ˆ we can get an estimate of x from x ˆ Given θ, ˆ = Ψθ. In the context of trajectory compression, x is the trajectory measured by an MSN node. The dimension of x is large. The MSN node computes y = Φx and transmits y to the server. The server can compute an estimated trajectory x ˆ by using y, Φ and Ψ, by solving (3). Note that the compression is lossy with 1 − m n represents both space savings and reduction in wireless transmission requirement.
The reconstruction error kˆ x − xk depends on a number of factors. The number of projections m must be large enough. A larger m generally reduces the reconstruction error but increases computation and transmission requirements at the sensor. The theory of CS shows that the number of projections m needed depends on the compressibility of the vector x in the basis Ψ. We say that a vector x is more compressible if its representation has fewer number of dominant elements, i.e. fewer non-zero elements with large magnitude. The theory of CS shows that a smaller m is needed if x is more compressible. We will propose an adaptive method to determine m on the MSN nodes in Section III. The choice of basis Ψ also affects the reconstruction error. A basic requirement is that x has to be compressible in the basis Ψ. In our previous work [5], we used standard bases for Ψ. In this paper, we show that learnt dictionary gives better performance. Another parameter that determines the achievable reconstruction error is the projection matrix Φ. The requirements on Φ to achieve low reconstruction error is expressed in terms of sensing matrix A = ΦΨ. Two requirements have been stated in the literature, in terms of Restricted Isometry Property (RIP) [19] and coherence [20]. We will discuss coherence. The mutual coherence µ(A) for the sensing matrix A is defined as µ(A) = max i
